College Algebra, Chapter 1, 1.3, Section 1.3, Problem 46

Find all real solutions of $\displaystyle z^2 + 5z + 3 = 0$.


$
\begin{equation}
\begin{aligned}

z^2 + 5z + 3 =& 0
&& \text{Given}
\\
\\
z^2 + 5z =& -3
&& \text{Subtract 3}
\\
\\
z^2 + 5z + \frac{25}{4} =& -3 + \frac{25}{4}
&& \text{Complete the square: add } \left( \frac{5}{2} \right)^2 = \frac{25}{4}
\\
\\
\left( z + \frac{5}{2} \right)^2 =& \frac{13}{4}
&& \text{Perfect square, take the LCD on the right side then simplify}
\\
\\
z + \frac{5}{2} =& \pm \sqrt{\frac{13}{4}}
&& \text{Take the square root}
\\
\\
z =& \frac{-5}{2} \pm \sqrt{\frac{13}{4}}
&& \text{Subtract } \frac{5}{2}
\\
\\
z =& \frac{-5}{2} \pm \sqrt{\frac{13}{4}}
&& \text{Subtract } \frac{5}{2}
\\
\\
z =& \frac{-5}{2} + \frac{\sqrt{13}}{2} \text{ and } z = \frac{-5}{2} - \frac{\sqrt{13}}{2}
&& \text{Solve for } z
\\
\\
z =& \frac{-5 + \sqrt{13}q}{2} \text{ and } z = \frac{-5 - \sqrt{13}}{2}
&& \text{Simplify}

\end{aligned}
\end{equation}
$